Nielsen-Schreier theorem

The set of Nielsen Schreier is a basic result of the combinatorial group theory , a partial area of mathematics which deals with discrete (usually infinite) groups. The sentence says that in a free group every subgroup is free. In addition to this qualitative statement, the quantitative version establishes a relationship between the index and the rank of a subgroup. This has the surprising consequence that a free group of rank has subgroups of any rank and even of (countably) infinite rank. ${\displaystyle r\geq 2}$ ${\displaystyle k\in \mathbb {N} }$

The theorem can be proved particularly elegantly and clearly with the help of algebraic-topological methods, by means of fundamental groups and superpositions of graphs.

Statement of the sentence

If there is a free group, then every subgroup of is also free. ${\displaystyle F}$ ${\displaystyle U}$ ${\displaystyle F}$

If the subgroup has a finite index, the following quantitative statement also applies:

If a free group is of rank and is a subgroup of finite index , then is free of rank . This is also true. ${\displaystyle F}$ ${\displaystyle r}$ ${\displaystyle U}$ ${\displaystyle n}$ ${\displaystyle U}$ ${\displaystyle 1+n(r-1)}$ ${\displaystyle r=\infty }$

proofs

The theorem can be proven with either algebraic or topological arguments. A purely algebraic proof can be found in the Robinson textbook given below . The topological proof is considered to be particularly elegant and will be outlined below. He cleverly uses the representation of free groups as fundamental groups of graphs and is a prime example of the fruitful interaction between algebra and topology.

Free groups as fundamental groups of graphs

Graph with spanning tree (black) and remaining edges (red). The latter freely generate the fundamental group of the graph. The generator belonging to the edge is shown in yellow as an example . ${\displaystyle s_{1}}$

Let be a connected graph . We realize this as a topological space, with each edge corresponding to a path between the adjacent corners. The crucial point now is that the fundamental group is a free group. ${\displaystyle \Gamma }$ ${\displaystyle \pi _{1}(\Gamma ,*)}$

To make this result explicit and thus also to prove it, one chooses a maximal tree , i.e. a tree that contains all vertices of . The remaining edges provide a basis for by choosing a path for each edge that runs from the base point in the tree to the edge , crosses this and then returns to the base point. (It is advisable to choose a corner of as the base point ; this is then automatically in every maximal tree .) The fact that the homotopy classes with form a basis of can be proven by means of combinatorial homotopy or by explicit construction of the universal overlay of the graph . ${\displaystyle T\subset \Gamma }$ ${\displaystyle \Gamma }$ ${\displaystyle S=\Gamma \setminus T}$ ${\displaystyle \pi _{1}(\Gamma ,*)}$ ${\displaystyle s\in S}$ ${\displaystyle w_{s}}$ ${\displaystyle *}$ ${\displaystyle T}$ ${\displaystyle s}$ ${\displaystyle T}$ ${\displaystyle \Gamma }$ ${\displaystyle T}$ ${\displaystyle [w_{s}]}$ ${\displaystyle s\in S}$ ${\displaystyle \pi _{1}(\Gamma ,*)}$ ${\displaystyle \Gamma }$

We can grasp this result quantitatively if there is a finite graph with corners and edges. It then has the Euler characteristic . Every maximal tree then contains exactly corners and edges, and in particular has the Euler characteristic . The edges and their number remain . The fundamental group is therefore a free group of rank . ${\displaystyle \Gamma }$ ${\displaystyle e}$ ${\displaystyle k}$ ${\displaystyle \chi (\Gamma )=e-k}$ ${\displaystyle T\subset \Gamma }$ ${\displaystyle e}$ ${\displaystyle e-1}$ ${\displaystyle \chi (T)=1}$ ${\displaystyle \Gamma \setminus T=\{x_{1},\dots ,x_{r}\}}$ ${\displaystyle r=k-e+1=1-\chi (\Gamma )}$ ${\displaystyle \pi _{1}(\Gamma ,*)}$ ${\displaystyle r=1-\chi (\Gamma )}$

Topological proof of the Nielsen-Schreier theorem

Qualitative version: Every free group can be represented as a fundamental group of a graph . Each subgroup has an overlay . The overlay space is then a graph again, so the group is free. ${\displaystyle F}$ ${\displaystyle \pi _{1}(\Gamma ,*)}$ ${\displaystyle \Gamma }$ ${\displaystyle U\subset F}$ ${\displaystyle {\tilde {\Gamma }}\to \Gamma }$ ${\displaystyle {\tilde {\Gamma }}}$ ${\displaystyle U=\pi _{1}({\tilde {\Gamma }},*)}$

Quantitative version: Every free group of finite rank can be represented as a fundamental group of a finite graph with Euler characteristics . Each subgroup of Index then has a -leaf overlay . The overlaying graph therefore has the Euler characteristic , and the group is therefore free of rank . ${\displaystyle F}$ ${\displaystyle r}$ ${\displaystyle \pi _{1}(\Gamma ,*)}$ ${\displaystyle \Gamma }$ ${\displaystyle \chi (\Gamma )=1-r}$ ${\displaystyle U\subset F}$ ${\displaystyle n}$ ${\displaystyle n}$ ${\displaystyle {\tilde {\Gamma }}\to \Gamma }$ ${\displaystyle {\tilde {\Gamma }}}$ ${\displaystyle \chi ({\tilde {\Gamma }})=n\chi (\Gamma )=n(1-r)}$ ${\displaystyle U=\pi _{1}({\tilde {\Gamma }},*)}$ ${\displaystyle 1-\chi ({\tilde {\Gamma }})=1-n(1-r)=1+n(r-1)}$

Geometric proof of Nielsen-Schreier's theorem

A group operation on an undirected graph, i.e. a homomorphism in the automorphism group of a graph, is called free if each group element different from the neutral element operates freely. The latter means that no node or edge is preserved in the operation. The geometric proof shows that a group is free if and only if it has a free group operation on a tree . Nielsen-Schreier's theorem is now a simple corollary, because this characterization of free groups is evidently carried over to subgroups.

Inferences

Subsets of whole numbers

For the rank is the trivial group consisting only of the neutral element, and the proposition of the proposition is empty. ${\displaystyle r=0}$ ${\displaystyle F}$

The first interesting statement we find in the rank . Here is the free Abelian group, and we find the classification of the subgroups of again: The trivial subgroup is free of rank , every other subgroup is free of the form of the index and itself again is free of the rank . This simple statement can also be proven without Nielsen-Schreier's theorem, but it shows what it is in the special case . ${\displaystyle r=1}$ ${\displaystyle F\cong \mathbb {Z} }$ ${\displaystyle \mathbb {Z} }$ ${\displaystyle \{0\}}$ ${\displaystyle 0}$ ${\displaystyle I\subset \mathbb {Z} }$ ${\displaystyle I=\langle n\rangle }$ ${\displaystyle n}$ ${\displaystyle 1}$ ${\displaystyle F\cong \mathbb {Z} }$

Subgroups of non-Abelian free groups

For a free group of rank it follows from the (quantitative) theorem of Nielsen-Schreier that there are free subsets of any finite rank. It is sufficient to show this for the group created by 2 elements , since this is contained in all free groups created by elements. If the two generators are mapped from to the generator of the cyclic group , a surjective group homomorphism is obtained from the defining property of the free group . According to the homomorphism theorem, is , that is, the subgroup has the index . According to the quantitative statement of the Nielsen-Schreier theorem, it is therefore isomorphic to the free group generated by elements. ${\displaystyle F}$ ${\displaystyle r\geq 2}$ ${\displaystyle F}$ ${\displaystyle F_{2}}$ ${\displaystyle r\geq 2}$ ${\displaystyle F_{2}}$ ${\displaystyle Z_{n}}$ ${\displaystyle \varphi :F_{2}\rightarrow \mathbb {Z} _{n}}$ ${\displaystyle F_{2}/\mathrm {ker} (\varphi )\cong \mathbb {Z} _{n}}$ ${\displaystyle \mathrm {ker} (\varphi )\subset F_{2}}$ ${\displaystyle n}$ ${\displaystyle 1+n}$

One can even construct into a subgroup of countably infinite rank. ${\displaystyle F_{2}}$

This amazing property is in contrast to free Abelian groups (where the rank of a subgroup is always less than or equal to the rank of the entire group) or vector spaces over a body (where the dimension of a subspace is always less than or equal to the dimension of the entire space).

Subgroups of finitely generated groups

Nielsen-Schreier's theorem deals initially only with free groups, but its quantitative version also has interesting consequences for any finite generated groups. If a group is finitely generated, starting from a family with elements , and is a subgroup of finite index , then also has a finite generating system with at most elements. ${\displaystyle G}$ ${\displaystyle r}$ ${\displaystyle G}$ ${\displaystyle H\subset G}$ ${\displaystyle n}$ ${\displaystyle H}$ ${\displaystyle 1+n(r-1)}$

As in the case of free groups, one must generally expect that a subgroup needs more producers than the entire group . ${\displaystyle H\subset G}$ ${\displaystyle G}$

history

The theorem is named after the mathematicians Jakob Nielsen and Otto Schreier and generalizes a theorem by Richard Dedekind that subsets of free Abelian groups are free Abelian groups, in the non-Abelian case. It was proved by Nielsen in 1921, but initially only for free subgroups of finite rank (finitely generated free subgroups). Schreier was able to lift this restriction in 1927 and generalize the theorem to any free groups. Max Dehn recognized the relationship to algebraic topology and, following the obituary for Dehn by Ruth Moufang and Wilhelm Magnus, was the first to give a topological proof of Nielsen-Schreier's theorem (unpublished). A representation of the proof with the help of graphs is also given by Otto Schreier in his treatise from 1927 (whereby he calls the graph a subgroup picture and regards it as an extension of Dehn's group picture from 1910). Further topological evidence comes from Reinhold Baer and Friedrich Levi and Jean-Pierre Serre . Kurt Reidemeister presented the connection of free groups with geometric topology in his textbook on combinatorial topology in 1932.

literature

• DL Johnson,: Topics in the Theory of Group Presentations, London Mathematical Society lecture note series, 42, Cambridge University Press, 1980 ISBN 978-0-521-23108-4 .
• Wilhelm Magnus, Abraham Karrass, Donald Solitar: Combinatorial Group Theory, 2nd edition, Dover Publications 1976.
• John Stillwell: Classical Topology and Combinatorial Group Theory, Graduate Texts in Mathematics, 72, Springer-Verlag, 2nd edition 1993.

See also

• The subgroup set of Kurosch generalizes the Nielsen-Schreier theorem on free products .

Web links

• NCatlab

Individual evidence

1. ^ DJS Robinson : A Course in the Theory of Groups , Springer-Verlag 1996, ISBN 978-1-4612-6443-9 , sentence 6.1.1: The Nielsen-Schreier-Theorem
2. ↑ Clara Löh: Geometric Group Theory , Springer-Verlag 2017, ISBN 978-3-319-72253-5 , Theorem 4.2.1 and Corollary 4.2.8
3. ^ DL Johnson, Topics in the Theory of Group Presentations, London Mathematical Society lecture note series, 42, Cambridge University Press 1980, p. 9
4. ↑ Nielsen, Om regning med ikke-kommutative factors og dens anvendelse i gruppeteorien, Math. Tidsskrift B, 1921, pp. 78-94
5. ↑ Otto Schreier : The subgroups of the free groups . In: Treatises from the Mathematical Seminar of the University of Hamburg . 5, 1927, pp. 161-183. doi : 10.1007 / BF02952517 .
6. ↑ See Wilhelm Magnus , Moufang, Ruth : Max Dehn zum Gedächtnis . In: Mathematical Annals . 127, No. 1, 1954, pp. 215-227. doi : 10.1007 / BF01361121 . SUB Göttingen , see p. 222
7. ↑ Schreier, 1927, p. 163, p. 180ff
8. ↑ Baer, ​​Levi, Free Products and their Subgroups, Compositio Mathematica, Volume 3, 1936, pp. 391–398
9. ^ Serre, Groupes discretes, Paris 1970
10. ↑ Kurt Reidemeister : Introduction to combinatorial topology . Wissenschaftliche Buchgesellschaft, Darmstadt 1972 (reprint of the original from 1932).